What is the fastest way to do a cut extrude / inward extrude?
Extrude the face you don't want to move then RMB to leave the duplicated edges in place.  Now select the face you want to move and you get the behaviour you want, except there is an edge loop that has been left behind.  Use Alt click to convert the selection to vertices, then Shift + Alt + RMB click the edge loop.  Translate the selection and you're done.